1. Gaussian Elimination

2. We use Gaussian elimination to solve systems of equations with 3 or more variables Our goal is to get the system reduced so that we can use substitution to solve. We do so by reducing the matrix to this form:
These can be any number
Ones across the diagonal and zeros in the left hand bottom corner.

3. Acceptable row operations:
Add two rows together
Multiply a row by a non zero number
Swap two rows
Multiply a row by a number and then add it to another row

4. Solve Using Gaussian Elimination
x+5y+3z = 10 -y+3z = 21 y–2z = 15 Step 1: Write the equations in a matrix of coefficients

5. We now have a The correct form!
Step 2: Use row operations to change the matrix to an augmented matrix
R2+ R3->R3
This -1 needs to be a 1
This 1 needs to be a 0
(-1)R2->R2
We now have a The correct form!

6. Step 3: Transform the matrix back into equations x+5y+3z = 10 y-3z = -21 z=36

7. Step 4: Solve using substitution x+5y+3z = 10 y-3z = -21 z=36